Gomory-Hu Trees

The network flow problem was first introduced by Ford Fulkerson who introduced the basic concept of flow, cut, etc. and provided the main tool, the maximum-flow minimum-cut theorem. Ford and Fulkerson wrote about the flow between two special points, the source and the sink. Mayeda then took up the multi-terminal problem, where flows are considered between all pairs of nodes in a network. In the discussed paper of Gomory and Hu it is continued with the multi-terminal problem, giving results on realizability, analysis and synthesis.

The Geometric Maximum Traveling Salesman Problem

The classical Traveling Salesman Problem asks for a tour through a given set of vertices such that the total distance of this tour is minimized. Instead of minimizing the total length, one can also asks for a maximum total tour length. It can be shown that such an optimal tour can be computed in time O(n) if distances are determined by the Manhatten metric, while it is NP-hard under Euclidean distances.

Algorithms for Ham-Sandwich Cuts

Given disjoint sets P_1, P_2, ..., P_d in R^d with n points in total, a ham sandwich cut is a hyperplane that simultaneously bisects the P_i. There is an algorithm for finding ham-sandwich cuts in every dimension d>1. When d=2, the algorithm is optimal, having complexity O(n). For dimension d>2, the bound on the running time is proportional to the worst-case time needed for constructing a level in an arrangement of n hyperplanes in dimension d-1.

Generalizing Ham-Sandwich Cuts to Equitable Subdivisions

Given gn red points and gm blue points in the plane in general position, there exists an equitable subdivision of the plane into g disjoint convex polygons, each of which contains n red points and m blue points.This is a generalization of the Ham Sandwich Theorem for the plane. There is also an efficient algorithm for constructing such equitable subdivisions.